Fourier Series Calculator

This free Fourier series calculator is exclusively designed to calculate the Fourier series of the given periodic function. Now, we have decided to commence with some basic theory!

What is Fourier Series?

In mathematics,

"The expansion of the periodic function in terms of infinite sums of sines and cosines is known as Fourier series."

Fourier Series Formula:

Take a look at the given formula that shows the periodic function f(x) in the interval \(-L\le \:x\le \:L\:\)

$$ f\left(x\right)=a_0+\sum _{n=1}^{\infty \:}a_n\cdot \cos \left(\frac{n\pi x}{L}\right)+\sum _{n=1}^{\infty \:}b_n\cdot \sin \left(\frac{n\pi x}{L}\right) $$

Where ;

$$ a_0=\frac{1}{2L}\cdot \int _{-L}^Lf\left(x\right)dx $$

$$ a_n=\frac{1}{L}\cdot \int _{-L}^Lf\left(x\right)\cos \left(\frac{n\pi x}{L}\right)dx,\:\quad \:n>0 $$

$$ b_n=\frac{1}{L}\cdot \int _{-L}^Lf\left(x\right)\sin \left(\frac{n\pi x}{L}\right)dx,\:\quad \:n>0 $$

With the help of the Fourier coefficients calculator, you can easily find values against these coefficients.

How Fourier Series is Calculated?

To determine the Fourier series of a function given may be a hectic and lengthy practice. That is why we have programmed our free fourier series coefficients calculator to determine the results instantly and precisely. But to understand the proper usage of Fourier series, let us solve a couple of examples.

Example # 01: Calculate fourier series of the function given below:

$$ f\left( x \right) = L - x on - L \le x \le L $$

Solution:

As, $$ f\left( x \right) = L - x $$ $$ f\left( -x \right) = -(L - x) $$

$$ f\left( x \right) = -f\left( x \right) $$

The given function is odd. Now, determining the coefficients as follows:

$$ {a_0} = \frac{1}{{2L}}\int_{{\, - L}}^{{\,L}}{{f\left( x \right)\,dx}} $$

$$ {a_0} = \frac{1}{{2L}}\int_{{\, - L}}^{{\,L}}{{L - x\,dx}} $$

$$ {a_0} = 2L $$   (click integral calculator for step by step calculations)

As we know that for an odd function, a_{n} is 0. Determining the value of b_{n} as follows:

$$ \begin{align*}{B_{\,n}} &= \frac{1}{L}\int_{{\, - L}}^{{\,L}}{{f\left( x \right)\sin \left( {\frac{{n\,\pi x}}{L}} \right)\,dx}} = \frac{1}{L}\int_{{\, - L}}^{{\,L}}{{\left( {L - x} \right)\sin \left( {\frac{{n\,\pi x}}{L}} \right)\,dx}}\\ & = \frac{1}{L}\left. {\left( { - \frac{L}{{{n^2}{\pi ^2}}}} \right)\left[ {L\sin \left( {\frac{{n\,\pi x}}{L}} \right) - n\pi \left( {x - L} \right)\cos \left( {\frac{{n\,\pi x}}{L}} \right)} \right]} \right|_{ - L}^L\\ & = \frac{1}{L}\left[ {\frac{{{L^2}}}{{{n^2}{\pi ^2}}}\left( {2n\pi \cos \left( {n\pi } \right) - 2\sin \left( {n\pi } \right)} \right)} \right] = \frac{{2L{{\left( { - 1} \right)}^n}}}{{n\pi }}\hspace{0.25in}\hspace{0.25in}n = 1,2,3, \ldots \end{align*} $$

(click integral calculator for step by step calculations) In this case, a_{0} is not zero but a_{n} is 0. So, the fourier series is given as:

$$ f\left(x\right)=2 L +\sum _{n=1}^{\infty \:}0\cdot \cos \left(\frac{n\pi x}{L}\right)+\sum _{n=1}^{\infty \:}\frac{2 \left(-1\right)^{n}}{n}\cdot \sin \left(\frac{n\pi x}{L}\right) $$

$$ f\left(x\right)=L + \sum_{n=1}^{\infty} \frac{2 \left(-1\right)^{n} \sin{\left(n x \right)}}{n} $$

Even here, a fourier coefficient calculator helps you to do particular calculations.

How Fourier Series Calculator Works?

Whenever you come across complex functions, our free online fourier series calculator is here to help you out in determining accurate results. You will get a proper scenario of the calculations by using our calculator.

Input:

• First, write your function in the drop down list
• After this, select the variable w.r t which you need to determine the Fourier series expansion
• Input the lower and upper limits
• Click ‘calculate’

Output: The Fourier expansion calculator calculates:

• Fourier series of the function given
• Fourier coefficients of the function f: a_{0}, a_{n}, and b_{n}
• Step by step calculations involved in the process

References:

From the source of Wikipedia: Convergence, Fourier series on a square, Hilbert space interpretation, Properties, Riemann–Lebesgue lemma, Riemann–Lebesgue lemma.